Explanation

The F-test in regression analysis is used to test linear restrictions on the coefficients. The key requirement is that the null hypothesis must be expressed as linear combinations of the regression coefficients.

Let's analyze each hypothesis:

I. β₁ = 1

• This can be written as: β₁ - 1 = 0
• This is a linear restriction (linear in β₁)
• ✅ Can be tested using F-test

II. β₂² = 1

• This involves a squared term: β₂²
• This is not a linear restriction
• ❌ Cannot be tested using F-test

III. β₃ = -2β₂

• This can be written as: β₃ + 2β₂ = 0
• This is a linear combination of β₃ and β₂
• ✅ Can be tested using F-test

IV. β₁β₂ = 0

• This involves a product of two coefficients
• This is not a linear restriction
• ❌ Cannot be tested using F-test

Therefore, only hypotheses I and III can be tested using an F-test.

Key Concept: The F-test requires that the null hypothesis consists of linear restrictions on the regression coefficients. Any hypothesis involving non-linear operations (squares, products, etc.) cannot be tested using the standard F-test.